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<H3>compute stress/atom command 
</H3>
<P><B>Syntax:</B>
</P>
<PRE>compute ID group-ID stress/atom keyword ... 
</PRE>
<UL><LI>ID, group-ID are documented in <A HREF = "compute.html">compute</A> command
<LI>stress/atom = style name of this compute command
<LI>zero or more keywords may be appended
<LI>keyword = <I>ke</I> or <I>pair</I> or <I>bond</I> or <I>angle</I> or <I>dihedral</I> or <I>improper</I> or <I>kspace</I> or <I>fix</I> or <I>virial</I> 
</UL>
<P><B>Examples:</B>
</P>
<PRE>compute 1 mobile stress/atom
compute 1 all stress/atom pair bond 
</PRE>
<P><B>Description:</B>
</P>
<P>Define a computation that computes the symmetric per-atom stress
tensor for each atom in a group.  The tensor for each atom has 6
components and is stored as a 6-element vector in the following order:
xx, yy, zz, xy, xz, yz.  See the <A HREF = "compute_pressure.html">compute
pressure</A> command if you want the stress tensor
(pressure) of the entire system.
</P>
<P>The stress tensor for atom <I>I</I> is given by the following formula,
where <I>a</I> and <I>b</I> take on values x,y,z to generate the 6 components of
the symmetric tensor:
</P>
<CENTER><IMG SRC = "Eqs/stress_tensor.jpg">
</CENTER>
<P>The first term is a kinetic energy contribution for atom <I>I</I>.  The
second term is a pairwise energy contribution where <I>n</I> loops over the
<I>Np</I> neighbors of atom <I>I</I>, <I>r1</I> and <I>r2</I> are the positions of the 2
atoms in the pairwise interaction, and <I>F1</I> and <I>F2</I> are the forces on
the 2 atoms resulting from the pairwise interaction.  The third term
is a bond contribution of similar form for the <I>Nb</I> bonds which atom
<I>I</I> is part of.  There are similar terms for the <I>Na</I> angle, <I>Nd</I>
dihedral, and <I>Ni</I> improper interactions atom <I>I</I> is part of.  There
is also a term for the KSpace contribution from long-range Coulombic
interactions, if defined.  Finally, there is a term for the <I>Nf</I>
<A HREF = "fix.html">fixes</A> that apply internal constraint forces to atom <I>I</I>.
Currently, only the <A HREF = "fix_shake.html">fix shake</A> and <A HREF = "fix_rigid.html">fix
rigid</A> commands contribute to this term.
</P>
<P>As the coefficients in the formula imply, a virial contribution
produced by a small set of atoms (e.g. 4 atoms in a dihedral or 3
atoms in a Tersoff 3-body interaction) is assigned in equal portions
to each atom in the set.  E.g. 1/4 of the dihedral virial to each of
the 4 atoms, or 1/3 of the fix virial due to SHAKE constraints applied
to atoms in a a water molecule via the <A HREF = "fix_shake.html">fix shake</A>
command.
</P>
<P>If no extra keywords are listed, all of the terms in this formula are
included in the per-atom stress tensor.  If any extra keywords are
listed, only those terms are summed to compute the tensor.  The
<I>virial</I> keyword means include all terms except the kinetic energy
<I>ke</I>.
</P>
<P>Note that the stress for each atom is due to its interaction with all
other atoms in the simulation, not just with other atoms in the group.
</P>
<P>The <A HREF = "dihedral_charmm.html">dihedral_style charmm</A> style calculates
pairwise interactions between 1-4 atoms.  The virial contribution of
these terms is included in the pair virial, not the dihedral virial.
</P>
<P>The KSpace contribution is calculated using the method in
<A HREF = "#Heyes">(Heyes)</A> for the Ewald method and by the methodology described
in <A HREF = "#Sirk">(Sirk)</A> for PPPM.  The choice of KSpace solver is specified
by the <A HREF = "kspace_style.html">kspace_style pppm</A> command.  Note that for
PPPM, the calcluation requires 6 extra FFTs each timestep that
per-atom stress is calculated.  Thus it can significantly increase the
cost of the PPPM calculation if it is needed on a large fraction of
the simulation timesteps.
</P>
<P>Note that as defined in the formula, per-atom stress is the negative
of the per-atom pressure tensor.  It is also really a stress*volume
formulation, meaning the computed quantity is in units of
pressure*volume.  It would need to be divided by a per-atom volume to
have units of stress (pressure), but an individual atom's volume is
not well defined or easy to compute in a deformed solid or a liquid.
Thus, if the diagonal components of the per-atom stress tensor are
summed for all atoms in the system and the sum is divided by dV, where
d = dimension and V is the volume of the system, the result should be
-P, where P is the total pressure of the system.
</P>
<P>These lines in an input script for a 3d system should yield that
result.  I.e. the last 2 columns of thermo output will be the same:
</P>
<PRE>compute		peratom all stress/atom
compute		p all reduce sum c_peratom[1] c_peratom[2] c_peratom[3]
variable	press equal -(c_p[1]+c_p[2]+c_p[3])/(3*vol)
thermo_style	custom step temp etotal press v_press 
</PRE>
<P><B>Output info:</B>
</P>
<P>This compute calculates a per-atom array with 6 columns, which can be
accessed by indices 1-6 by any command that uses per-atom values from
a compute as input.  See <A HREF = "Section_howto.html#howto_15">Section_howto
15</A> for an overview of LAMMPS output
options.
</P>
<P>The per-atom array values will be in pressure*volume
<A HREF = "units.html">units</A> as discussed above.
</P>
<P><B>Restrictions:</B> none
</P>
<P><B>Related commands:</B>
</P>
<P><A HREF = "compute_pe.html">compute pe</A>, <A HREF = "compute_pressure.html">compute pressure</A>
</P>
<P><B>Default:</B> none
</P>
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<A NAME = "Heyes"></A>

<P><B>(Heyes)</B> Heyes, Phys Rev B 49, 755 (1994),
</P>
<A NAME = "Sirk"></A>

<P><B>(Sirk)</B> Sirk, Moore, Brown, J Chem Phys, 138, 064505 (2013).
</P>
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